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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval and let $f \\colon I \\to \\R$.\n\n1. The function $f$ is of *class* $C^0$ on $I$ if it is [[continuous|0P27W9RW0G]] on $I$.\n2. Let $k \\ge 1$ be an integer. The function $f$ is of *class* $C^k$ on $I$ if the [[higher derivatives|084NGSTWM7]] $f', \\ldots, f^{(k)}$ exist on $I$. Moreover $f^{(k)}$ must be continuous on $I$.\n3. The function $f$ is of *class* $C^\\infty$ on $I$ if it is of class $C^k$ for every $k \\ge 0$.\n\nThe set of functions of class $C^k$ on $I$ is written $C^k(I)$.\n\n## Remarks\n\nThe classes are nested: $C^\\infty(I) \\subseteq \\cdots \\subseteq C^2(I) \\subseteq C^1(I) \\subseteq C^0(I)$.\nThe class $C^1$ is the class of [[continuously differentiable functions|06SKCHDQT5]], and $C^\\infty$ is the class of [[smooth functions|0BE3QDH73A]].\nThe class $C^\\omega$ of real [[analytic functions|0JD5ZNW0F4]] lies inside $C^\\infty$.\nFor functions of several variables, $C^k$ is defined by continuous partial derivatives up to order $k$; see [[the characterisation by derivatives|0TCBZMEYGN]].\n\nExample: $f(x) = x \\abs{x}$ on $\\R$ is of class $C^1$.\nIts derivative is $f'(x) = 2 \\abs{x}$, which is continuous.\nIt is not of class $C^2$, because $2\\abs{x}$ is not differentiable at $0$.\nMore generally, $x^k \\abs{x}$ is of class $C^k$ but not of class $C^{k+1}$.\n\nNon-example: $g(x) = x^2 \\sin(1/x)$ for $x \\ne 0$, with $g(0) = 0$, is differentiable on $\\R$ but not of class $C^1$.\nIts derivative is $g'(x) = 2x \\sin(1/x) - \\cos(1/x)$ for $x \\ne 0$ and $g'(0) = 0$, and $g'$ has no limit at $0$.\n",
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